Probing decoherence in molecular 4f qubits

Abstract

We probe herein the fundamental factors that induce decoherence in ensembles of molecular magnetic materials. This is done by pulse Electron Paramagnetic Resonance measurements at X-band (∼9.6 GHz) on single crystals of Gd@Y(trensal) at 0.5, 10^-1, 10^-2 and 10^-3% doping levels, using Hahn echo, partial refocusing and CPMG sequences. The phase memory time, T _m, obtained by the Hahn echo sequence at X-band is compared to the one previously determined at higher frequency/magnetic field (∼240 GHz). The combined information from these experiments allows to gain insight into the contributions to decoherence originating from various relaxation mechanisms such as spin-lattice relaxation, electron and nuclear spin diffusion and instantaneous diffusion. We show that while at high magnetic fields T _m is limited by spin-lattice relaxation seemingly attributed to a direct process, at lower fields the limiting factor is spectral diffusion. At X-band, for Gd@Y(trensal) we determine a T _m in the range 1-12 μs, at 5 K, depending on the magnetic field and concentration of Gd(trensal) in the isostructural diamagnetic host Y(trensal). Importantly, Gd@Y(trensal) displays measurable coherence at temperatures above liquid nitrogen ones, with 125 K being the upper limit. At the lowest dilution level of 10^-3% and under dynamic decoupling conditions, the ratio of T _m versus the time it takes to implement a quantum gate, T _G, reaches the order of 10⁴, in the example of a single qubit π-rotation, which corresponds to an upper limit of gate fidelity of the order of 99.99%, reaching thus the lower limit of qubit figure of merit required for implementations in quantum information technologies.

Fig. 1. Molecular structure of Gd(trensal) (a) seen along the threefold symmetry axis and (b) seen from the side. Colour code: C, gray; H, white; N, nitrogen; O, oxygen; Gd, cyan.

Fig. 2. EDFS (blue) of 0.5% with B₀ ∥ C₃ (top) and of 10⁻³% with B₀ ⊥ C₃ (bottom) at 5.5 K, compared to simulations (black) using previously determined CF parameters (Table S3†).

Fig. 3. Field dependence of T₁ and T_m for 0.5% with B₀ ∥ C₃ (top) and for 10⁻³% with B₀ ⊥ C₃ (bottom) at 5.5 K.

Fig. 4. Temperature dependence of T₁ and T_m obtained by EPR measurements of 0.5% and 10⁻³% with B₀ ∥ C₃ and B₀ ⊥ C₃. The dashed lines are linear fits to log(T₁) vs. log(T) as described in the main text.

Fig. 5. Inverse concentration dependence of T_m with B₀ ⊥ C₃ at 5 (10⁻¹% and 10⁻²%) and 5.5 (0.5% and 10⁻³%) K. The solid lines are guides for the eye.

Fig. 6. T _m of 10⁻³% with B₀ ⊥ C₃ at 5.5 K measured via a Hahn echo sequence with variable length of the refocusing pulse at 4367, 3817, 3249 and 2917 Gauss, showing no angle dependence. The solid lines are guides for the eye.

Fig. 7. T _m of 10⁻³% with B₀ ⊥ C₃ at 5.5 K as a function of the number of refocusing CPMG pulses at 4367, 3817, 3249 and 2917 Gauss. The solid lines are guides for the eye.

Fig. 8. (Top): Schematic of the transient nutation pulse sequence and associated Rabi oscillations of 10⁻³% with B₀ ⊥ C₃ at 5.5 K for the central (4 ↔ 5)_⊥ transition. (Bottom): Rabi frequencies plotted against the relative magnitude of the magnetic field component of the microwave pulse (B1).